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%%文档的题目、作者与日期
%\author{王立庆（2020级数学与应用数学1班） }
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{常微分方程期末练习}
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\date{2023 年 12 月 12 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %第1题
求下述齐次线性微分方程的通解：
\begin{enumerate}[label={(\arabic*)}]
\item  $y''+9y'+20y=0$. 
\item  $y''-2y'+y=0$. 
\item  $y'''-y=0$. 
\item  $y^{(4)}-y''=0$. 
\item  $y^{(4)}+y=0$. 
\item  $y'''-y''-y'+y=0$. 
\end{enumerate}

\vspace{0.2cm}

\vspace{8cm}

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\item  %第2题
求下列方程满足给定初值条件的解：
\begin{enumerate}[label={(\arabic*)}]
\item  $y''-3y'+2y=0$, $y(0)=2$, $y'(0)=-3$. 
\item  $y''+4y'+4y=0$, $y(2)=4$, $y'(2)=0$. 
\end{enumerate}

\vspace{0.2cm}

\vspace{0.2cm}

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\item  %第3题
考虑常微分方程 $ x^2\frac{d^2y}{dx^2} - x\frac{dy}{dx} +y = 0$. 
\begin{enumerate}[label={(\arabic*)}]
\item  设变量代换 $x=e^t$, 将原方程化为关于 $y$ 与 $t$ 的常微分方程。
\item  求解该方程。
\end{enumerate}

\vspace{0.2cm}

\vspace{8cm}

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\item  %第4题
求下述非齐次线性微分方程的通解：
\begin{enumerate}[label={(\arabic*)}]
\item  $y''-7y'+12y=24$. 
\item  $y''+2y'+5y=26e^{2x}$. 
\item  $y''-2y'=8x+5\cos x$. 
\end{enumerate}
\vspace{0.2cm}


\vspace{0.2cm}

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\item  %第5题
考虑关于未知函数 $y(x)$ 的常微分方程 $$\frac{dy}{dx} = \frac{6}{(1-2x)(1-3y)}. $$
\begin{enumerate}[label={(\arabic*)}]
\item  设变量代换 $t=2x, u=3y$, 将原方程化为关于未知函数 $u(t)$ 的常微分方程。
\item   设初值条件为 $y(0)=0$, 求这个方程在原点附近的幂级数解的前三项（求 $t,t^2,t^3$ 的系数），并求这个幂级数解的收敛半径。
\end{enumerate}

\vspace{0.2cm}


\vspace{8cm}

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\item  %第6题
考虑平面动力系统
\begin{eqnarray*}
\left\{\begin{array}{rcl}
\frac{dx}{dt} &=& 3x+y, \\
\frac{dy}{dt} &=& -x+y.
\end{array}\right.
\end{eqnarray*}
\begin{enumerate}[label={(\arabic*)}]
\item  求出通解。
\item  判断奇点的类型和稳定性。
\item  求出轨线族的方程。
\item  画出相图。
\end{enumerate}

\vspace{0.2cm}

\vspace{0.2cm}

\newpage
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\item  %第7题
考虑微分方程组
\begin{eqnarray*}
\left\{\begin{array}{rcl}
\frac{dx}{dt} &=& y, \\
\frac{dy}{dt} &=& -y-\sin x.
\end{array}\right.
\end{eqnarray*}

\begin{enumerate}[label={(\arabic*)}]
\item  求这个平面动力系统的平衡点。
\item  考虑李雅普诺夫函数 $V(x,y)=y^2+2-2\cos x$, 验证零解的稳定性。
\end{enumerate}

\vspace{0.2cm}


\vspace{8cm}

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\item  %第8题
考虑常微分方程 $x^2y''+xy'+(x^2-4)y=0$. 
\begin{enumerate}[label={(\arabic*)}]
\item  判断 $x_0=0$ 是方程的常点、正则奇点还是非正则奇点。
\item  求广义幂级数解的指标根，以及系数的递推关系式。
\end{enumerate}

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\end{enumerate}


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\end{document}

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